Implementability and equity in production economies with unequal skills

Abstract

In this paper, we study the problem of fair allocation in production economies where agents have different preferences and unequal production skills. We characterize the equal income Walrasian solution and the proportional solution using axioms of equity and a certain form of implementability developed by Yamada and Yoshihara (Int J Game Theory 36:85–106, 2007).

Appendix

In this appendix, we show the independence of the axioms in the theorems. First we provide solutions satisfying all the axioms in Theorem 1 except:

• (1) PE. Let $e' \in \mathcal {E}$ be such that for all $\varvec{z}\in \textit{EI}(e')$, $c_{i^*}>0$ for some $i^* \in N$ and $R_{i^*} \not = R_{j}$ for all $j \not = i^*$. Define $S^1$ as follows. For all $e \in \mathcal {E} \backslash \{e'\}$, $\varvec{z}\in S^1(e)$ if $\varvec{z}\in \textit{EI}(e)$; for $e'$, $\varvec{z}\in S^1(e')$ if $z_{i^*}=(l_{i^*}', c_{i^*}' - \epsilon )$ for a sufficiently small $\epsilon >0$, $\varvec{z}_{-i} = \varvec{z}_{-i}'$, where $\varvec{z}' \in \textit{EI}(e')$.

• (2) SPI. Consider the Egalitarian Equivalent solution EE considered by Fleurbaey and Maniquet (1999, Definition 15).

• (3) IUS. Consider the $\tilde{u}$ budget equivalent solution $BE^{\tilde{u}}$ (other than $\textit{EI}$) introduced by Fleurbaey and Maniquet (1996).⁸

• (4) EREL. Define $S^2$ as follows. Let $\overline{B}(s_{i},w, b_{i})=\{z' \in [0,\overline{l}] \times \mathbb {R}_{+}| c' \le ws_{i}l' + b_{i}\}$. For all $e \in \mathcal {E}$, $\varvec{z}\in S^2(e)$ if $\varvec{z}\in P(e)$ and there exists $w \in W(\varvec{z}, e)$ such that for all $i \in N$,

  $$\begin{aligned} z_{i} \in \mathop {\text{ argmax }}_{\tilde{z} \in \overline{B}(s_{i},w, b_{i})}u_{i}(\tilde{z}), \end{aligned}$$

  where $b_{i} \not = b_{j}$ for all $i,j \in N$.

• (5) ND. Consider the proportional solution.

Next we present examples of solutions satisfying all the axioms in Theorem 2 but:

• (1) PE. Let $e' \in \mathcal {E}$ be such that for all $\varvec{z}\in \textit{PR}(e')$, $c_{i^*}>0$ for some $i^* \in N$ and $R_{i^*} \not = R_{j}$ for all $j \not = i^*$. Define $S^3$ as follows. For all $e \in \mathcal {E} \backslash \{e'\}$, $\varvec{z}\in S^3(e)$ if $\varvec{z}\in \textit{PR}(e)$; for $e'$, $\varvec{z}\in S^3(e')$ if $z_{i^*}=(l_{i^*}', c_{i^*}' - \epsilon )$ for a sufficiently small $\epsilon >0$, $\varvec{z}_{-i} = \varvec{z}_{-i}'$, where $\varvec{z}' \in \textit{PR}(e')$.

• (2) SPI. Let $S^4$ be defined as follows. For all $e \in \mathcal {E}$, $\varvec{z}\in S^4(e)$ if $\varvec{z}$ is in the Egalitarian Equivalent solution EE(e) and for all $i, j \in N$ such that $R_{i} = R_{j}$ and $s_{i}=s_{j}$, $z_{i}=z_{j}$.

• (3) IUS. Define $S^5$ as follows. For all $e \in \mathcal {E}$, $\varvec{z}\in S^5(e)$ if $\varvec{z}\in BE^{\tilde{u}}$ (other than $\textit{EI}$) and for all $i, j \in N$ such that $R_{i} = R_{j}$ and $s_{i}=s_{j}$, $z_{i}=z_{j}$.

• (4) NRZE. Consider the equal income Walrasian solution.

• (5) NDL. Consider $S^6$ such that for all $e \in \mathcal {E}$, $S^6(e) \subseteq \textit{PR}(e)$ and $|S^6(e)|=1$.